SOCN -- Graduate school in Systems, Optimization, Control and Networks

Graduate school courses scheduled in the academic year

2007-2008

Fall 2007

Spring 2008

"MECHATRONICS : DYNAMICS OF ELECTROMECHANICAL AND PIEZOELECTRIC SYSTEMS, WITH APPLICATIONS IN VIBRATION CONTROL"
Lecturer : André PREUMONT (Université Libre de Bruxelles, Belgium) CANCELLED

"ROBUST TRACKING AND DISTURBANCE REJECTION "
Lecturer : Alberto ISIDORI (University of Roma, Italy)

"CONSTRUCTIVE LMI APPROACHES FOR ANTI-WINDUP COMPENSATOR DESIGN FOR SYSTEMS SUBJECT TO SATURATION"
Lecturer : Sophie TARBOURIECH (LAAS-CNRS, Toulouse, France)

Detailed contents

1. OPTIMIZATION ALGORITHMS ON MATRIX MANIFOLDS

Lecturers : P.-A. ABSIL (Université Catholique de Louvain, Belgium) and R. SEPULCHRE (Université de Liège, Belgium)

Overview

The course will offer an introduction to the theory and applications of numerical optimization on manifolds. This area of computational mathematics deals with the analysis and design
of algorithms for optimizing a real-valued function whose domain, instead of being the classical Euclidean space Rn, is in general a non-Euclidean space endowed with a differentiable structure.
This includes real-valued functions on smooth surfaces of Rn, and more generally on sets that can be “smoothly” parameterized. Optimization techniques on manifolds find applications in several areas, including the eigenvalue problem, blind source separation (via independent component analysis), pose estimation in computer vision, and model reduction of dynamical systems.

Aim of the course

• Motivate why manifold theory provides natural foundations for the development of matrix algorithms for several major computational problems and why the resulting numerical algorithms can be highly competitive.
• Provide the students with the necessary background in differential geometry instrumental to algorithmic developments.
• Develop classical optimization algorithms on matrix manifolds: steepest-descent, Newton, conjugate gradients, trust-region methods.
• Guide the student through the concrete calculations that turn an abstract geometric algorithm into a numerical implementation.
• Illustrate the many problems from linear algebra, advanced signal processing, and statistical analysis that can be recast as optimization problems on matrix manifolds.

Contents

1. Motivating examples: Eigenvalue problem, independent component analysis, computer vision, dynamical systems. Manifolds, submanifolds, quotient manifolds.
2. First-order geometry: Tangent vectors, Riemannian metric, gradient vector fields, gradient flows.
3. First-order algorithms: Retractions, seepest-descent methods, line-search strategies. Optimization on the orthogonal group and independent component analysis.
4. Second-order geometry and Newton’s method: Newton’s method in Rn, affine connections, Riemannian connection, parallel translation, geodesics, Newton’s method on manifolds.
5. Models and trust-region methods. Optimization of the Rayleigh quotient on the sphere and on the Grassmann manifold.
6. Vector transport, approximate Newton methods, conjugate gradients. Applications.

Prerequisites

Basic knowledge of matrix algebra and real analysis. No specific background in differential geometry or numerical optimization will be assumed.

Supporting material
The course will be primarily based on the book “Optimization algorithmson matrix manifolds”, coauthored by P.-A. Absil, R. Mahony and R. Sepulchre, to appear.

Evaluation procedure

The students will design and implement an algorithm for independent component analysis and apply it to image processing or gene expression analysis problems. The
task will be split into homeworks.

Dates : October 01, 04, 08, 11, 15, 18, 2007.

Schedule : 9h15 - 12h15

This course will take place at CESAME, Bâtiment Euler, 4, av. G. Lemaître, 1348 Louvain-la-Neuve

2. NUMERICAL METHODS FOR NONLINEAR OPTIMAL CONTROL PROBLEMS

Lecturer : M. DIEHL (K.U.Leuven, Belgium)

The course develops numerical solution strategies for optimization problems with underlying differential equation models. After a brief overview over sequential and simultaneous approaches to optimal control, we focus on the latter class, which solve both the model equations and the optimality conditions in a "one-shot' approach. Here, exploitation of sparsity in the linear solvers is of crucial importance. Special topics include the treatment of distributed inequalities by interior point methods, periodic problems. Finally, we discuss real world applications from chemical and mechanical engineering.

Contents

1. Optimal Control: Introduction and Overview
2. Dynamic Programming and Indirect Approaches (Pontryagin)
3. Direct Transcription Methods (ODE Sensititivity, Single Shooting)
4. Direct Collocation, Treatment of Sparsity
5. Direct Multiple Shooting
6. Applications and Extensions: Multi-Stage Processes, Periodicity

Prerequisites

Participants are assumed to have a knowledge of linear algebra and calculus. Basic knowledge in optimisation and numerical simulation of dynamic systems is advantageous.

Supporting material

Notes will be distributed during the course.

Related references

J. T. Betts: Practical Methods for Optimal Control Using Nonlinear Programming, SIAM, Philadelphia, 2001
Bryson, A.E. and Ho, Y.-C.: Applied Optimal Control, Wiley, New York, 1975

Dates : November 26, 29 and December 03, 06, 07, 2007

Schedule : 9h15 - 12 h45

Location :

26 november : room 02.58

29 november, 03,06, 07 december : room 00.62

This course will take place at the Katholieke Universiteit Leuven,
Department Elektrotechniek ESAT, Kasteelpark Arenberg 1, 3001 Heverlee

LINKS OF INTEREST:

K.U. Leuven Optimization in Engineering Center, OPTEC:

http://homes.esat.kuleuven.be/~optec/agenda.php

Join the OPTEC announcement list:

http://listserv.cc.kuleuven.ac.be/archives/optec.html

SLIDES AND ARTICLES

http://homes.esat.kuleuven.be/~optec/events/courses/GraduateSchool/

3. MECHATRONICS : DYNAMICS OF ELECTROMECHANICAL AND PIEZOELECTRIC SYSTEMS,

WITH APPLICATIONS IN VIBRATION CONTROL

Lecturer : A. PREUMONT (Université Libre de Bruxelles, Belgium)

CANCELLED !!!

Contents

1) Lagrangian dynamics of mechanical systems :
Kinetic state functions, Virtual work principle, D'Alembert's principle, Hamilton's principle, Lagrange's equation, gyroscopic systems, Jacobi integral, Rayleigh-Ritz method.
2) Lagrangian dynamic of electrical network :

Constitutive equations of circuits elements, Hamilton's principle (charge formulation and flux linkage formulation), Lagrange's equation (charge formulation and flux linkage formulation)
3) Lagrangian dynamics of electromechanical systems :
Constitutive equations for transducers (movable-plate capacitor, movable-core inductor, moving-coil transducer)

Hamilton's principle and Lagrange's equation (charge formulation and flux linkage formulation)
Examples : loudspeaker, microphone, electrodynamic isolator, geophone, magnetic suspension.
Control examples: sky-hook damper, magnetic levitation.
Self-sensing.
4) Piezoelectric systems :
Constitutive equations of piezoelectric transducers, electromechanical coupling factor, structure with a piezoelectric transducer.
Piezoelectric material (constitutive equations, coenergy density function).
Hamilton's principle.
Rosen's transformer.
5) Piezoelectric laminates :
Beam actuator, Hamilton's principle, piezoelectric loads, laminar sensor, modal actuator and sensor.
Beam with collocated actuator-sensor pair, pole-zero pattern, modal truncation.
Kirchhoff plate, multi-layer laminate.
6) Active and passive damping with piezoelectric transducers :
Active strut, Integral Force Feedback, voltage vs. current control, modal coordinates.
Damping via resistive shunting and inductive shunting.
Self-sensing.

4. ROBUST TRACKING AND DISTURBANCE REJECTION

Lecturer : A. Isidori (University of Roma, Italy)

Contents

1. Limit sets and steady state behavior of nonlinear systems

These lectures review the concepts of limit set of a point and

the (less common) concept of limit set of a set. The latter

provides a firm conceptual foundation to the extension to

nonlinear system of the familiar notion of steady-state behavior

and steady-state response. Several illustrative examples are

provided (2 hours)

2. A review of certain stability properties

This lecture reviews the notion of stability of a compact

attractor, its role in the steady-state analysis, and the

associated converse Lyapunov theorem (1 hour).

3. The generalized servomechanism problem for linear systems

These lectures review some basic results on the problem of

(robust) asymptotic tracking and/or disturbance rejection (with

exogenous inputs modelled as solutions of a give differential

equation) for linear systems. The case of exogenous inputs

modelled by uncertain exosystems is also covered (4 hours).

4. The generalized servomechanism problem for nonlinear systems

and the nonlinear internal model principle

These lecture introduce the servomechanism problem for nonlinear

systems. In particular, it is shown how the concepts of steady

state response leads to a rather general formulation of the

so-called internal model principle. Moreover, a general paradigm

for the design of a regulator is presented. (2 hours)

5. Nonlinear observers and the design of internal models

These lectures show how the theory of nonlinear is instrumental in

the design of an internal model. Two approaches to the design of

nonlinear observers are discussed: the high-gain observer and the

so-called nonlinear Luenberger observer (3 hours)

6. Robust stabilization of attractors via small-gain theorem

These lectures review the concepts of input-to-state stability (to

compact attractors) and the associated small gain theorem. These

results are then used to complete the design of a controller that

solves the generalized servomechanism problem (2 hours)

7. Examples and open problems

Dates : April 15 ,16, 18, 21, 23, 25, 2008

Schedule : 9h15 - 12 h45

This course will take place at CESAME, Bâtiment Euler, 4, av. G. Lemaître, 1348 Louvain-la-Neuve

5. CONSTRUCTIVE LMI APPROACHES FOR ANTI-WINDUP COMPENSATOR DESIGN FOR SYSTEMS SUBJECT TO SATURATION

Lecturer : S. TARBOURIECH (LAAS-CNRS, Toulouse, France)

Physical, safety or technological constraints induce that the control actuators can neither provide unlimited amplitude signals nor unlimited speed of reaction. The control problems of combat aircraft prototypes and satellite launchers offer interesting examples of the difficulties due to these major constraints. Neglecting actuator saturations on both amplitude and dynamics can be source of undesirable or even catastrophic behavior for the closed-loop system (such as loosing closed-loop stability).
Such actuator saturations have also been blamed as one of several unfortunate mishaps leading to the 1986 Chernobyl nuclear power plant disaster. For these reasons, the study of the control problem (its structure, performance and stability analysis) for systems subject to both amplitude and rate actuator saturations as typical input constraints has received the attention of many researchers in the last years.

Contents

1. General introduction
2. Context of stability
3. Approximating the saturation term
4. Introduction on anti-windup strategy
5. Solution via static anti-windup (1 loop)
6. Solution via static anti-windup (2 loops)
7. Solution via dynamical anti-windup
8. Concluding remarks and perspectives

Overview

The anti-windup approach consists of taking into account the effect of saturations in a second step after a previous design performed disregarding the saturation terms. The idea is then to introduce control modifications in order to recover, as much as possible, the performance induced by a previous design carried out on the basis of the unsaturated system. During this course, we will present some pertinent elements related to these anti-windup techniques and to illustrate their potentialities on some applications. In this context, LMI techniques have allowed to obtain constructive conditions to design the suitable anti-windup loops satisfying performance and stability requirements.

Prerequisites

Participants are assumed to have some background in linear algebra, Lyapunov theory, linear systems and control theory.

References

– J-M. Biannic, S. Tarbouriech, D. Farret, A practical approach to performance analysis of saturated systems with application to fighter aircraft flight
controllers, 5th IFAC Symposium ROCOND, Toulouse, France, July, 2006.
– Y.Y. Cao, Z. Lin, and D.G. Ward. An anti-windup approach to enlarging domain of attraction for linear systems subject to actuator saturation. IEEE
Transactions on Automatic Control, 47(1):140-145, 2002.
– J.M. Gomes da Silva Jr. and S. Tarbouriech, Anti-windup design with guaranteed region of stability: an LMI-based approach, IEEE Transactions on
Automatic Control, 50:1, pp.106-111, 2005.
– G. Grimm, J. Hatfield, I. Postlethwaite, A.R. Teel, M.C. Turner, and L. Zaccarian. Anti-windup for stable linear systems with input saturation: an LMI based synthesis. IEEE Transactions on Automatic Control, 48(9):1509-1525, 2003.
– P. Hippe. Windup in control. Its effects and their prevention. AIC, Springer, Germany, 2006.
– S. Tarbouriech, G. Garcia, A.H. Glattfelder (Eds.), Advanced strategies in control systems with input and output constraints, LNCIS, vol.346, Springer Verlag, 2007.
– A.R. Teel, A nonlinear small gain theorem for the analysis of control systems with saturation, IEEE Transactions on Automatatic Control, 41, pp.1256-1270, 1996.
– M. Turner and L. Zaccarian (Editors). Special issue: Anti-windup. International Journal of System Science, 37(2):65-139, 2006.

Dates : May 5, 6, 7, 13, 14, 15, 2008

Schedule : 9h15 - 12h15

This course will take place at CESAME, Bâtiment Euler, 4, av. G. Lemaître, 1348 Louvain-la-Neuve

Papers :

Registration

You can register electronically by filling in the following form via the web :

http://www.inma.ucl.ac.be/graduate/graduate_registration.html

If you have problems with this, please contact Lydia DE BOECK

The admission is free for doctoral students and participants from Belgian academic institutions. Other participants are requested to pay a registration fee of EURO 500,- per course but a waiver can be obtained under special conditions (contact the secretariat).
Payment can be made by bank transfer to the account n°310-0959001-48 with the mention "AUTO2866 ACTIVITES DIDACTIQUES"

Secretariat

Lydia DE BOECK
CESAME - Bât. Euler
4, av. G. Lemaître
1348 Louvain-la-Neuve (Belgium): Tél : 010/47 80 36; fax : 010/ 47 21 80; e-mail : lydia.deboeck@uclouvain.be; web site : http://www.inma.ucl.ac.be/graduate